multiplying by variables in inequalities

Hey Gurpinder,

Great question, and one that's pretty important. When using variables and inequalities, you need to consider whether the variable is negative...you don't multiply the variable by -1.

In your example, consider that the following two values of x both satisfy the inequality:

4/x < 1/3

Well, any negative value of x turns that whole term negative, so -1 would give us -4 < 1/3 ---> true

And any value of x that makes 4/x smaller then 4/12 (which is 1/3) also works, so x could be 18: 4/18 < 1/3 because 2/9 is less then 1/3.

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Now, algebraically, what you want to do is recognize that you don't know whether x is positive or negative. If it's positive, when performing the algebra you don't change the sign:

4/x < 1/3 becomes
4 < x/3
12 < x


But if x were negative, you would have to flip the sign. So:

4/x < 1/3 becomes

4 > x/3
12 > x (BUT remember - this is only if x is negative, so 9, 10, or 11 don't work...this really just means that any negative number less than 12 (so all negative numbers) also satisfies the equation).



So...what's important is that you consider that there are negative values that also satisfy the inequality. This happens most often in Data Sufficiency, where often testing a number or two on the negative side helps to bear this out conceptually maybe even more than just using the algebra (which is admittedly a little awkward above).

Thanks Brian,

Always great hearing you!